> My understanding is that, in many ways, government agencies are more constitutionally protected speech-wise than private entities

That's government employees.

If an employee of a normal company goes to a bar after work and trash-talks their boss, they can get fired for it.

If an employee of the government does the same, they (probably) can't.

This only covers speech that's not the of the job, and only things that are of "public concern" whatever that means.